Complex Analysis - Maths KU

Remember: Arg(z) is in the interval
(−π, π].
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2.4 Special Power Functions 91
A. This implies that z1 �= −z2, since z1 is in A. Therefore, we conclude that
z1 = z2, and this proves that f is a one-to-one function on A.
The technique of Example 7 does not extend to the function z n , n>2.
For this reason we present an alternative approach to show that f(z) =z 2
is one-to-one on A, which can be modified to show that f(z) =z n , n>2,
is one-to-one on an appropriate domain. As in Example 7, we will prove
that f(z) =z 2 is one-to-one on A byshowing that if f(z1) =f(z2) for two
complex numbers z1 and z2 in A, then z1 = z2. Suppose that z1 and z2 are
in A, then we maywrite z1 = r1e iθ1 and z2 = r2e iθ2 with −π/2